furnace/extern/fftw/doc/html/1d-Real_002dodd-DFTs-_0028D...

<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- This manual is for FFTW
(version 3.3.10, 10 December 2020).

Copyright (C) 2003 Matteo Frigo.

Copyright (C) 2003 Massachusetts Institute of Technology.

Permission is granted to make and distribute verbatim copies of this
manual provided the copyright notice and this permission notice are
preserved on all copies.

Permission is granted to copy and distribute modified versions of this
manual under the conditions for verbatim copying, provided that the
entire resulting derived work is distributed under the terms of a
permission notice identical to this one.

Permission is granted to copy and distribute translations of this manual
into another language, under the above conditions for modified versions,
except that this permission notice may be stated in a translation
approved by the Free Software Foundation. -->
<!-- Created by GNU Texinfo 6.7, http://www.gnu.org/software/texinfo/ -->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>1d Real-odd DFTs (DSTs) (FFTW 3.3.10)</title>

<meta name="description" content="1d Real-odd DFTs (DSTs) (FFTW 3.3.10)">
<meta name="keywords" content="1d Real-odd DFTs (DSTs) (FFTW 3.3.10)">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<link href="index.html" rel="start" title="Top">
<link href="Concept-Index.html" rel="index" title="Concept Index">
<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="What-FFTW-Really-Computes.html" rel="up" title="What FFTW Really Computes">
<link href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html" rel="next" title="1d Discrete Hartley Transforms (DHTs)">
<link href="1d-Real_002deven-DFTs-_0028DCTs_0029.html" rel="prev" title="1d Real-even DFTs (DCTs)">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.indentedblock {margin-right: 0em}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
kbd {font-style: oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
span.nolinebreak {white-space: nowrap}
span.roman {font-family: initial; font-weight: normal}
span.sansserif {font-family: sans-serif; font-weight: normal}
ul.no-bullet {list-style: none}
-->
</style>


</head>

<body lang="en">
<span id="g_t1d-Real_002dodd-DFTs-_0028DSTs_0029"></span><div class="header">
<p>
Next: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html" accesskey="n" rel="next">1d Discrete Hartley Transforms (DHTs)</a>, Previous: <a href="1d-Real_002deven-DFTs-_0028DCTs_0029.html" accesskey="p" rel="prev">1d Real-even DFTs (DCTs)</a>, Up: <a href="What-FFTW-Really-Computes.html" accesskey="u" rel="up">What FFTW Really Computes</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<span id="g_t1d-Real_002dodd-DFTs-_0028DSTs_0029-1"></span><h4 class="subsection">4.8.4 1d Real-odd DFTs (DSTs)</h4>

<p>The Real-odd symmetry DFTs in FFTW are exactly equivalent to the unnormalized
forward (and backward) DFTs as defined above, where the input array
<em>X</em> of length <em>N</em> is purely real and is also <em>odd</em> symmetry.  In
this case, the output is odd symmetry and purely imaginary.
<span id="index-real_002dodd-DFT-1"></span>
<span id="index-RODFT-1"></span>
</p>

<span id="index-RODFT00"></span>
<p>For the case of <code>RODFT00</code>, this odd symmetry means that
<i>X<sub>j</sub> = -X<sub>N-j</sub></i>,
where we take <em>X</em> to be periodic so that
<i>X<sub>N</sub> = X</i><sub>0</sub>.
Because of this redundancy, only the first <em>n</em> real numbers
starting at <em>j=1</em> are actually stored (the <em>j=0</em> element is
zero), where <em>N = 2(n+1)</em>.
</p>
<p>The proper definition of odd symmetry for <code>RODFT10</code>,
<code>RODFT01</code>, and <code>RODFT11</code> transforms is somewhat more intricate
because of the shifts by <em>1/2</em> of the input and/or output, although
the corresponding boundary conditions are given in <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html">Real even/odd DFTs (cosine/sine transforms)</a>.  Because of the odd symmetry, however,
the cosine terms in the DFT all cancel and the remaining sine terms are
written explicitly below.  This formulation often leads people to call
such a transform a <em>discrete sine transform</em> (DST), although it is
really just a special case of the DFT.
<span id="index-discrete-sine-transform-2"></span>
<span id="index-DST-2"></span>
</p>

<p>In each of the definitions below, we transform a real array <em>X</em> of
length <em>n</em> to a real array <em>Y</em> of length <em>n</em>:
</p>
<span id="RODFT00-_0028DST_002dI_0029"></span><h4 class="subsubheading">RODFT00 (DST-I)</h4>
<span id="index-RODFT00-1"></span>
<p>An <code>RODFT00</code> transform (type-I DST) in FFTW is defined by:
<center><img src="equation-rodft00.png" align="top">.</center>
</p>
<span id="RODFT10-_0028DST_002dII_0029"></span><h4 class="subsubheading">RODFT10 (DST-II)</h4>
<span id="index-RODFT10"></span>
<p>An <code>RODFT10</code> transform (type-II DST) in FFTW is defined by:
<center><img src="equation-rodft10.png" align="top">.</center>
</p>
<span id="RODFT01-_0028DST_002dIII_0029"></span><h4 class="subsubheading">RODFT01 (DST-III)</h4>
<span id="index-RODFT01"></span>
<p>An <code>RODFT01</code> transform (type-III DST) in FFTW is defined by:
<center><img src="equation-rodft01.png" align="top">.</center>
In the case of <em>n=1</em>, this reduces to
<i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
</p>
<span id="RODFT11-_0028DST_002dIV_0029"></span><h4 class="subsubheading">RODFT11 (DST-IV)</h4>
<span id="index-RODFT11"></span>
<p>An <code>RODFT11</code> transform (type-IV DST) in FFTW is defined by:
<center><img src="equation-rodft11.png" align="top">.</center>
</p>
<span id="Inverses-and-Normalization-1"></span><h4 class="subsubheading">Inverses and Normalization</h4>

<p>These definitions correspond directly to the unnormalized DFTs used
elsewhere in FFTW (hence the factors of <em>2</em> in front of the
summations).  The unnormalized inverse of <code>RODFT00</code> is
<code>RODFT00</code>, of <code>RODFT10</code> is <code>RODFT01</code> and vice versa, and
of <code>RODFT11</code> is <code>RODFT11</code>.  Each unnormalized inverse results
in the original array multiplied by <em>N</em>, where <em>N</em> is the
<em>logical</em> DFT size.  For <code>RODFT00</code>, <em>N=2(n+1)</em>;
otherwise, <em>N=2n</em>.
<span id="index-normalization-11"></span>
</p>

<p>In defining the discrete sine transform, some authors also include
additional factors of
&radic;2
(or its inverse) multiplying selected inputs and/or outputs.  This is a
mostly cosmetic change that makes the transform orthogonal, but
sacrifices the direct equivalence to an antisymmetric DFT.
</p>
<hr>
<div class="header">
<p>
Next: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html" accesskey="n" rel="next">1d Discrete Hartley Transforms (DHTs)</a>, Previous: <a href="1d-Real_002deven-DFTs-_0028DCTs_0029.html" accesskey="p" rel="prev">1d Real-even DFTs (DCTs)</a>, Up: <a href="What-FFTW-Really-Computes.html" accesskey="u" rel="up">What FFTW Really Computes</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
</div>


</body>
</html>